Sharpening the Arithmetic of Active Management

I challenge William F. Sharpe's famous equality that "before costs, the return on the average actively managed dollar will equal the return on the average passively managed dollar." This equality is based on the implicit assumption that the market portfolio never changes, which does not hold in the real world because new shares are issued, others are repurchased, and indexes are reconstituted - so even "passive" investors must regularly trade. Therefore, active managers can be worth positive fees in aggregate, allowing them to play an important economic role: helping allocate resources efficiently. Passive investing also plays a useful economic role: creating low-cost access to markets.